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Last updated on 15 September 2025
Both addition and subtraction are considered as two basic operations in math. Here, we will explore how these operations apply to complex numbers. A complex number is expressed in the form of a+ib, where a, b are real numbers. Let’s learn more about them and how to add or subtract them in this article.
In mathematics, a complex number is denoted as z. Since it is a combination of real and imaginary numbers, a complex numbers is represented as z = a + ib. Here, the equation indicates an imaginary unit, while a and b are the real and imaginary parts, respectively. The value of i is (√-1).
While adding or subtracting complex numbers, the real and imaginary parts are combined. When the terms are combined, we can perform either addition or subtraction.
Before directly performing addition or subtraction of complex numbers, follow the steps below for better understanding:
We perform addition by combining the real parts and imaginary parts of two numbers separately. For example, if z1 = a + ib and z2 =c + id are two complex numbers. The addition of these two numbers can be performed as:
z1 + z2 = (a + ib) + (c + id) = (a + c) + i(b + d)
Let z1 = 3 + 4i and z2 = 2 + 5i.
Then, z1 + z2 = (3 + 4i) + (2 + 5i)
When we add these two complex numbers, first we must add the real parts:
3 + 2 = 5
Next, we can add the imaginary parts:
4 + 5 = 9
Thus, z1 + z2 = (3 + 4i) + (2 + 5i) = 5 +9i
When subtracting complex numbers, the real and imaginary parts are subtracted individually.
z1 - z2 = (a + ib) - (c + id) = (a - c) + i(b - d)
For example, subtract (6 + 5i) - (4 + 3i)
Step 1: Subtract real parts: 6 - 4 = 2.
Step 2: Subtract imaginary parts: 5 - 3 = 2i.
Result: 2 + 2i
Thus, (6 + 5i) - (4 + 3i) = 2 + 2i
Key properties of complex number addition and subtraction include:
Closure property: The closure property states that the result we obtain from the addition and subtraction of complex numbers is also a complex number. For example, if we subtract (7 + 5i) - (4 + 3i): We get 3 + 2i, which is also a complex number.
Associative property: This property applies only to the addition of three complex numbers. The subtraction of complex numbers does not show the associative property, but the addition exhibits the property. If we add three complex numbers, then:
(z1+ z2) + z3 = z1+ (z2 + z3)
Commutative property: The commutative property states that only the addition of two or more complex numbers is commutative. For example, if z1 and z2 are complex numbers, then z1 + z2 = z2 + z1
Additive identity: Additive identity is a number that, when added to a complex number, does not change its value. For any complex number z = a + bi, the additive identity is z + 0 = (a + bi) + 0 = a + bi
Additive inverse: -z is the additive inverse of a complex number, z. Therefore, z + (-z) = 0.
Understanding the properties and concepts of complex numbers is useful in various fields. These include engineering, physics, mathematics, and computer science. Some practical, real-world applications of complex numbers are listed below:
While doing addition and subtraction of complex numbers, students often make mistakes in calculations. They often forget to consider the real part of the complex numbers. Here are some common errors and solutions to avoid them:
Add (2 + 3i) and (1 + 4i)
3 + 7i
To add two complex numbers, we can use the formula:
(a + ib) + (c + id) = (a + c) + i(b + d)
Where, a = 2
b = 3
c = 1
d = 4
Now we can substitute the values and identify the real and imaginary parts:
(2 + 3i) + (1 + 4i) = (2 + 1) + i(3 + 4) = 3 + 7i
Thus, the sum of (2 + i3) and (1 + i4) is 3 + 7i.
Subtract (6 + 5i) from (3 + 2i)
-3 - 3i
To subtract two complex numbers, we can use the formula:
(a + ib) - (c + id) = (3 + 2i) - (6 + 5i)
Now we can substitute the values and identify the real and imaginary parts:
(a - c) + i (b - d) = (3 - 6) + i(2 -5)
(3 - 6) = -3
(2 - 5) = -3
Thus, the difference when (6 + 5i) is subtracted from (3 + 2i) is -3 - 3i
Add (7 - 3i) and (-4 + 8i)
3 + 5i
To add two complex numbers, we can use the formula:
(a + ib) + (c + id) = (a + c) + i(b + d)
Where, a = 7
b = -3
c = -4
d = 8
Now we can substitute the values and identify the real and imaginary parts:
(a + c) + i (b + d) = (7 + (-4)) + i(-3 + 8)
Add the real parts:
7 + (-4) = 3
Add the imaginary parts:
-3 + 8 = 5
Thus, (7 - 3i) + (-4 + 8i) = 3 + 5i
Subtract (-6 + 5i) and (3- 4i)
-9 + 9i
To subtract two complex numbers, we can use the formula:
(a + ib) - (c + id) = (a - c) + i(b - d)
Where, a = - 6
b = 5
c = 3
d = -4
Now we can substitute the values:
(-6 + 5i) - (3 + (-4) i) = (-6 - 3) + i(5 - (-4)
Simplify the real parts:
-6 - 3= -9
Simplify the imaginary parts:
5 - (-4) = 5 + 4 = 9
Thus, the final result is -9 + 9i
(-6 + 5i) - (3 - 4i) = -9 + 9i
Add the two complex numbers z = 4 - 8i and w = 5 - 6i
9 - 14i
To add two complex numbers, we can use the formula:
(a + ib) + (c + id) = (a + c) + i(b + d)
Where, a = 4
b = -8
c = 5
d = -6
Now, we can substitute the values:
(4 - 8i) + (5 - 6i) = (4 + 5) + i(-8 + (-6))
= 9 + i (-14)
= 9 - 14 i
Thus, (4 - 8i) + (5 - 6i) = 9 - 14i
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.